Equations of Lines• Vertical line x = #• Horizontal line y = #• Slope, y-intercept y=mx+b • Standard Form Ax+By = C using
intercepts • Standard Form using magic
COORDINATE PLANE
Parts of a plane1. X-axis2. Y-axis3. Origin4. Quadrants I-IV
X-axis
Y-axis
Origin ( 0 , 0 )
QUAD IQUAD II
QUAD III QUAD IV
SLOPE ReviewSlope is the ratio of the vertical rise to the horizontal run between any two points on a line. Usually referred to as the rise over run. Slope triangle between two
points. Notice that the slope triangle can be drawn two different ways.
Rise is -10 because we went down
Run is -6 because we went to the left
3
5
6
10
iscasethisinslopeThe
Rise is 10 because we went up
Run is 6 because we went to the right
3
5
6
10iscasethisinslopeThe
SLOPE-INTERCEPT FORM OF A LINEThe slope intercept form of a line is y = mx + b“m” represents the slope “b” represents the y-intercept.
♥When an equation is in slope-intercept form the
“y” is always on one side by itself.
♥If a line is not in slope-intercept form, then we must solve for “y” to get it there.
Write the equation of the line with a y-intercept of 3
and a slope of 2 Then…. graph it.
y = 2x + 3
First stepSecond step
Third step – draw the line
IN SLOPE-INTERCEPT NOT IN SLOPE-INTERCEPT
y = 3x – 5 y – x = 10
y = -2x + 10 2y – 8 = 6x
y = -.5x – 2 y + 4 = 2x
Put y – x = 10 into slope-intercept form
Add x to both sides and would get y = x + 10
Put 2y – 8 = 6x into slope-intercept form.
Add 8 to both sides then divide by 2 and would get y = 3x + 4
Put y + 4 = 2x into slope-intercept form.
Subtract 4 from both sides and would get y = 2x – 4.
The x-intercept is point where graph touches (or crosses) the x-axis.
The y-intercept is point where graph touches (or crosses) the y-axis.
1. To find x-intercepts, let y be zero and solve the equation for x.
2. To find y-intercepts, let x be zero and solve the equation for y.
Finding Interceptsof an Equation
X AND Y INTERCEPTSThe x-intercept is the x-coordinate of a point where the graph crosses the x-axis.
The y-intercept is the y-coordinate of a point where the graph crosses the y-axis.
The x-intercept would be 4 and is located at the point (4, 0).
The y-intercept is 3 and is located at the point (0, 3).
Finding the interceptsx + 5y = 10
find the x-interceptx + 5(0) = 10x = 10 find the y-intercept
0 + 5y = 105y = 10y = 2
The coordinates are (10, 0) and (0, 2)
1. Find your x-intercept: Let y = 0-2x + 3(0) = 12x = -6; (-6, 0)
2. Find your y-intercept:Let x = 0-2(0) + 3y = 12y = 4; (0, 4)
3. Graph both points and draw a line through them.
Graphing with intercepts:-2x + 3y = 12
Write the equation in Standard Form:Ax + By = C
y = 2x + 3Get your variables on one side of the equation and the constant on the other.
-2x + y = 3
You’re not done…. the coefficient of x must be a positive integer.
2x – y = -3
Write the equation in Standard Form:Ax + By = C
y = 2/5x - 3Get your variables on one side of the equation and the constant on the other.
-2/5x + y = -3
Check the coefficient of x. Guess we’re not done yet.Multiply the equation by -52x – 5y = 15
Write the equation in Standard Form:Ax + By = C
y = 2/5x - 3OR….You recognize that the coefficient of x needs work: Multiply the equation by 5.
5y = 2x - 3
Move the variables to one side:-2x + 5y = -15Multiply by -12x – 5y = 15
Graphing a line from Standard Form Using Magic
Slope = -A B
x - intercept =C A
y - intercept =C B
Ax + By = C
2x – y = -3
For example:
Slope = -A B
x - intercept =C A
y - intercept =C B
Slope = -2 = 2
-1x - intercept =-3 2
y - intercept =-3 = 3 -1
Knowing this can set up your equation to graph in EITHER slope-intercept form or graphing by intercepts!!
Here’s your cheat sheet!♥ If the equation is in STANDARD FORM
(Ax + By = C), graph using the intercepts or your magic formulas.
♥ If the equation is in SLOPE-INTERCEPT FORM (y = mx + b), graph using slope and intercept or a t-table (whichever is easier for you).
♥ If the equation is in NEITHER form, rewrite the equation in the form you like the best!